In graph theory, cubicity is a graph invariant defined to be the smallest dimension such that a graph can be realized as an intersection graph of unit cubes in Euclidean space.
Cubicity was introduced by Fred S. Roberts in 1969 along with a related invariant called boxicity that considers the smallest dimension needed to represent a graph as an intersection graph of axis-parallel rectangles in Euclidean space.
can be realized as an intersection graph of axis-parallel unit cubes in
-dimensional Euclidean space.
[2] The cubicity of a graph is closely related to the boxicity of a graph, denoted
The definition of boxicity is essentially the same as cubicity, except in terms of using axis-parallel rectangles instead of cubes.
Since a cube is a special case of a rectangle, the cubicity of a graph is always an upper bound for the boxicity of a graph.
In the other direction, it can be shown that for any graph
is the ceiling function, i.e., the smallest integer greater than or equal to